General Nonlinear Dynamical Systems Research Articles

Decomposition and equivalence of general nonlinear dynamical control systems

AbstractProjection and equivalence concepts have been widely studied in the literature. In the present paper two nonlinear systems with the same number of inputs, but not the same number of state variables, are considered. Under mild assumptions, necessary and sufficient conditions are given for the existence of a submersion such that the higher dimensional system projects locally onto the other one. The solution to this problem has relevant applications, for instance in robotics. It includes as a special case the equivalence of nonlinear systems with no particular structure.

Adaptive exchange protocol for multi-agent communication in augmented reality system

Augmented Reality (AR) is one of the modern ways of providing users with different types of information. In applications based on this technology, the main influence on the subjective assessment of product quality is the speed of interaction between the system and the user and between users of the system. The main purpose of the work series are creation and modification for algorithms to assessing and improving the quality of AR services. This paper describes and justifies an adaptive communication protocol for multi-agent interaction. We consider a general nonlinear dynamic system with introducing feedback control which is based on measurements under almost arbitrary noise. In practice, this control is realized as a superposition of neural networks with the estimation of the result of mutual additional learning based on the system identification method (M.K.Campi and E.Weyer’s LSCR is used). The prototype is built for a system with a distributed server architecture and multiagent behavior of both clients and servers in the system.

PiP-X: Online feedback motion planning/replanning in dynamic environments using invariant funnels

Computing kinodynamically feasible motion plans and repairing them on-the-fly as the environment changes is a challenging, yet relevant problem in robot navigation. We propose an online single-query sampling-based motion re-planning algorithm using finite-time invariant sets, commonly referred to as “ funnels”. We combine concepts from nonlinear systems analysis, sampling-based techniques, and graph-search methods to create a single framework that enables feedback motion re-planning for any general nonlinear dynamical system in dynamic workspaces. A volumetric network of funnels is constructed in the configuration space using sampling-based methods and invariant set theory; and an optimal sequencing of funnels from robot configuration to a desired goal region is then determined by computing the shortest-path subgraph (tree) in the network. Analyzing and formally quantifying the stability of trajectories using Lyapunov level-sets ensures kinodynamic feasibility and guaranteed set-invariance of the solution paths. Though not required, our method is capable of using a pre-computed library of motion primitives to speedup online computation of controllable motion plans that are volumetric in nature. We introduce a novel directed-graph data structure to represent the funnel-network and its inter-sequencibility; helping us leverage discrete graph-based incremental search to quickly rewire feasible and controllable motion plans on-the-fly in response to changes in the environment. We validate our approach on a simulated cart-pole, car-like robot, and 6DOF quadrotor platform in a variety of scenarios within a maze and a random forest environment. Using Monte Carlo methods, we evaluate the performance in terms of algorithm success, length of traversed trajectory, and runtime.

Scalable Projection-Based Reduced-Order Models for Large Multiscale Fluid Systems

Although projection-based reduced-order models (PROMs) have existed for decades, they have rarely been applied to large, nonlinear, multiscale, and multi-physics systems due to the complexity of effectively implementing such methods. Advances in hyper-reduction have enabled the scalable computation of PROMs for general nonlinear dynamical systems. Further, the recent model-form-preserving least squares with variable transformation method has proven capable of generating stable PROMs for extremely stiff multiphysics problems. In this work, we formulate a PROM framework combining these methodologies and demonstrate that robust, accurate, and cost-effective PROMs can be realized for complex nonreacting and reacting compressible flows. Along with an open-source toolchain for hyper-reduction sample mesh generation from extremely large data sets, this represents an end-to-end effort to assess the applicability of PROMs to large-scale, multiphysics problems of engineering interest. We examine practical considerations for implementing hyper-reduction methods and their effect on memory consumption, load balancing, and interprocessor communications. These considerations produce accurate PROMs that are three to four orders of magnitude more computationally efficient than the full-order model in recreating transonic flow over a cavity and reacting flow in a rocket combustor. Guidelines for data preparation, sample mesh construction, and online PROM solution which promote robust simulations are also provided.

Finite/fixed-time event-triggered aperiodic intermittent control for nonlinear systems

In this paper, some new event-triggered aperiodic intermittent control methods are proposed to stabilize nonlinear systems in finite-time (FT) and fixed-time (FXT). Firstly, two event-triggered aperiodic intermittent control schemes are designed for general nonlinear dynamical systems by employing a class of auxiliary functions. Under the proposed control schemes and using Lyapunov stability theory, some FT stabilization criteria are obtained. Secondly, to stabilize nonlinear dynamical systems in FXT, some improved event-triggered aperiodic intermittent control strategies are given and the corresponding stabilization criteria are also obtained. Moreover, as an application of these strategies, the consensus of nonlinear multi-agent systems (MASs) is investigated via designing node-based and edge-based event-triggered aperiodic intermittent control protocols, and some FXT consensus conditions are derived. Finally, the theoretical results are verified by two examples.

An efficient implementation of graph-based invariant set algorithm for constrained nonlinear dynamical systems

The graph-based invariant set (GIS) algorithm is a promising set-based technique for computing the largest (with respect to inclusion) control invariant set of general discrete-time nonlinear dynamical systems. However, like other invariant set algorithms for nonlinear systems, the GIS algorithm may require a lot of resources when computing the control invariant set. This limits its applicability to higher dimensional systems. In this work, we present an improved and efficient implementation of the GIS algorithm for general discrete-time controlled nonlinear systems. We first identify the bottlenecks through extensive analysis, and then provide remedial procedures to improve the implementation of the GIS algorithm. Specifically, we developed an adaptive subdivision scheme using a supervised machine learning-based algorithm to reduce the cell growth rate and parallelize the graph construction step. We extensively demonstrate the performance of the improved GIS algorithm using a numerical example and compare the result to that of the standard GIS algorithm. The results show that the adaptive subdivision and the parallelization improved the speed of the algorithm by about 8x and 3x respectively, that of the standard GIS algorithm.

Data-based modeling and identification for general nonlinear dynamical systems by the multidimensional Taylor network

PurposeThe present study is intended to develop an effective approach to the real-time modeling of general dynamic nonlinear systems based on the multidimensional Taylor network (MTN).Design/methodology/approachThe authors present a detailed explanation for modeling the general discrete nonlinear dynamic system by the MTN. The weight coefficients of the network can be obtained by sampling data learning. Specifically, the least square (LS) method is adopted herein due to its desirable real-time performance and robustness.FindingsCompared with the existing mainstream nonlinear time series analysis methods, the least square method-based multidimensional Taylor network (LSMTN) features its more desirable prediction accuracy and real-time performance. Model metric results confirm the satisfaction of modeling and identification for the generalized nonlinear system. In addition, the MTN is of simpler structure and lower computational complexity than neural networks.Research limitations/implicationsOnce models of general nonlinear dynamical systems are formulated based on MTNs and their weight coefficients are identified using the data from the systems of ecosystems, society, organizations, businesses or human behavior, the forecasting, optimizing and controlling of the systems can be further studied by means of the MTN analytical models.Practical implicationsMTNs can be used as controllers, identifiers, filters, predictors, compensators and equation solvers (solving nonlinear differential equations or approximating nonlinear functions) of the systems of ecosystems, society, organizations, businesses or human behavior.Social implicationsThe operating efficiency and benefits of social systems can be prominently enhanced, and their operating costs can be significantly reduced.Originality/valueNonlinear systems are typically impacted by a variety of factors, which makes it a challenge to build correct mathematical models for various tasks. As a result, existing modeling approaches necessitate a large number of limitations as preconditions, severely limiting their applicability. The proposed MTN methodology is believed to contribute much to the data-based modeling and identification of the general nonlinear dynamical system with no need for its prior knowledge.

Functional Bayesian Filter

We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). By applying the kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of the Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned online from measurement data in the form of an infinite impulse response (IIR) filter or recurrent network in the RKHS, with universal approximation property. Using a positive definite kernel function satisfying Mercer’s conditions to compute and evolve information quantities, the FBF exploits both the statistical and time-domain information about the signal, extracts higher-order moments, and preserves the properties of covariances without the ill effects due to conventional arithmetic operations. We apply this novel kernel adaptive filtering (KAF) to recurrent network training, chaotic time-series estimation and cooperative filtering using Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation results show FBF outperforms existing Kalman-based algorithms.

An Efficient Weighted Residual Time Integration Family

A new family of time integration methods is formulated. The recommended technique is useful and robust for the loads with large variations and the systems with nonlinear damping behavior. It is also applicable for the structures with lots of degrees of freedom, and can handle general nonlinear dynamic systems. By comparing the presented scheme with the fourth-order Runge–Kutta and the Newmark algorithms, it is concluded that the new strategy is more stable. The authors’ formulations have good results on amplitude decay and dispersion error analyses. Moreover, the family orders of accuracy are [Formula: see text] and [Formula: see text] for even and odd values of [Formula: see text], respectively. Findings demonstrate the superiority of the new family compared to explicit and implicit methods and dissipative and non-dissipative algorithms.

Explicit expression of stationary response probability density for nonlinear stochastic systems

Identifying the exactly or approximately explicit expression of the stationary response probability density for general nonlinear stochastic dynamical systems is of great significance in the fields of stochastic dynamics and control. Almost all the existing methods are devoted to determine the exact or approximate solution for specific values of system and excitation parameters. Herein, aimed at stochastic systems with polynomial nonlinearity and excited by Gaussian white noises, a novel method is proposed to identify the stationary response probability density which explicitly includes system and excitation parameters. The stationary probability density is first written as an exponential function according to the maximum entropy principle, the power of the exponential function is then expressed as a linear combination of prescribed nondimensional parameter clusters constituted by system and excitation parameters, and state variables, with the coefficients to be determined. The undetermined coefficients are derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov equation. The application and efficacy of the proposed method are illustrated by a typical numerical example.

Lyapunov-Based Real-Time and Iterative Adjustment of Deep Neural Networks

A real-time Deep Neural Network (DNN) adaptive control architecture is developed for general uncertain nonlinear dynamical systems to track a desired time-varying trajectory. A Lyapunov-based method is leveraged to develop adaptation laws for the output-layer weights of a DNN model in real-time while a data-driven supervised learning algorithm is used to update the inner-layer weights of the DNN. Specifically, the output-layer weights of the DNN are estimated using an unsupervised learning algorithm to provide responsiveness and guaranteed tracking performance with real-time feedback. The inner-layer weights of the DNN are trained with collected data sets to increase performance, and the adaptation laws are updated once a sufficient amount of data is collected. Building on the results in (Joshi and Chowdhary, 2019) and (Joshi et al., 2020), which focus on deep model reference adaptive control for linear systems with known drift dynamics and control effectiveness matrices, this letter considers general control-affine uncertain nonlinear systems. The real-time controller and adaptation laws enable the system to track a desired time-varying trajectory while compensating for the unknown drift dynamics and parameter uncertainties in the control effectiveness. A nonsmooth Lyapunov-based analysis is used to prove semi-global asymptotic tracking of the desired trajectory. Numerical simulation examples are included to validate the results, and the Levenberg-Marquardt algorithm is used to train the weights of the DNN.

Extremum seeking control with attenuated steady-state oscillations

We propose two perturbation-based extremum seeking control (ESC) schemes for general single input single output nonlinear dynamical systems, having structures similar to that of the classical ESC scheme. We propose novel adaptation laws for the excitation signal amplitudes in each scheme that drive the amplitudes to zero. The rates of decay for both the laws are governed by the gradient measures of the unknown reference-to-output equilibrium map We show that the proposed ESC schemes achieve practical asymptotic convergence to the extremum with a proper tuning of the parameters in the proposed schemes. As the extremum is reached, and the magnitudes of the gradient measures become small, the excitation signal amplitudes converge to zero. Thus, the proposed schemes ensure that the excitation signal is driven to zero as the system output converges to a neighborhood of the extremum and the steady-state oscillations about the extremum, typically observed for the classical ESC schemes, are attenuated. Simulation examples are included to illustrate the effectiveness of the proposed schemes.

New trends in modeling and control of hybrid systems

New trends in modeling and control of hybrid systems

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

SummaryMultiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved.

Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc

We introduce Sim.DiffProc, an R package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations (SDEs). It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms, Ito and Stratonovich. One of Sim.DiffProc key features is to implement the Monte Carlo method for the iterative evaluation and approximation of an interesting quantity at a fixed time on SDEs with parallel computing, on multiple processors on a single machine or a cluster of computers, which is an important tool to improve capacity and speed-up calculations. We also provide an easy-to-use interface for symbolic calculation and numerical approximation of the first and central second-order moments of SDEs (i.e., mean, variance and covariance), by solving a system of ordinary differential equations, which yields insights into the dynamics of stochastic systems. The final result object of Monte Carlo and moment equations can be derived and presented in terms of LATEX math expressions and visualized in terms of LATEX tables. Furthermore, we illustrate various features of the package by proposing a general bivariate nonlinear dynamic system of Haken-Zwanzig, driven by additive, linear and nonlinear multiplicative noises. In addition, we consider the particular case of a scalar SDE driven by three independent Wiener processes. The Monte Carlo simulation thereof is obtained through a transformation to a system of three equations. We also study some important applications of SDEs in different fields.

Newton-based extremum seeking: A second-order Lie bracket approximation approach

In this paper, we present novel multi-variable Newton-based extremum seeking systems, based on Lie bracket approximation methods. More precisely, we consider cost functions with an unknown mathematical description, but whose value can be measured on-line. We propose extremum seeking systems that approximate the Newton-based optimization law, by combining the on-line measurement of the cost with time-periodic excitation signals. The inversion of the Hessian matrix is avoided by introducing a first order dynamical system, whose output approximates the Newton step. This provides practical robustness with respect to ill-conditioned Hessian matrices. Semi-global stability properties of the proposed schemes are demonstrated both for static cost functions and for cost functions associated with a general non-linear dynamical system. The effectiveness of the approach is shown in simulations.

On existence and continuation of solutions of the state-dependent impulsive dynamical system with boundary constraints

The state-dependent impulsive dynamical system with boundary constraints is a kind of special but common system in nature. But because of the complexity of the geometry or topological structures of the impulsive surface, it is hard to determine when an event or an impulsive surface is reached. Therefore, a general state-dependent impulsive nonlinear dynamical system is rarely studied. This paper presents a class of state-dependent impulsive dynamical systems with boundary constraints. We obtain the existence and continuation of their viable solutions and provide sufficient conditions for the existence and uniqueness of the viable solutions to the system. Finally, two examples are given to illustrate the effectiveness of the results.

Stabilization and destabilization of nonlinear stochastic differential delay equations

ABSTRACTThis present paper deals with the problem of stabilization and destabilization of nonlinear stochastic differential delay equations. The key techniques used in this paper are the LaSalle-type stability theorems, the nonnegative semimartingale convergence theorem, and the law of large numbers for martingales. The proposed results can be applied to study the stabilization and destabilization of more general nonlinear stochastic dynamical time-delay systems. Computer simulations are presented to illustrate our theory.

Non-intrusive subdomain POD-TPWL for reservoir history matching

This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) for reservoir history matching through integrating domain decomposition (DD), proper orthogonal decomposition (POD), radial basis function (RBF) interpolation, and the trajectory piecewise linearization (TPWL). It is an efficient approach for model reduction and linearization of general non-linear time-dependent dynamical systems without accessing to the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain is saved and then partitioned into subdomains. From the local sequence of snapshots over each subdomain, a number of local basis vectors is formed using POD, and then the RBF interpolation is used to estimate the derivative matrices for each subdomain. Finally, those derivative matrices are substituted into a POD-TPWL algorithm to form a reduced-order linear model in each subdomain. This reduced-order linear model makes the implementation of the adjoint easy and results in an efficient adjoint-based parameter estimation procedure. Comparisons with the classic finite-difference-based history matching show that our proposed subdomain POD-TPWL approach is obtaining comparable results. The number of full-order model simulations required is roughly 2–3 times the number of uncertain parameters. Using different background parameter realizations, our approach efficiently generates an ensemble of calibrated models without additional full-order model simulations.

Ultra Rapid Data Assimilation Based on Ensemble Filters

The goal of this work is to analyse and study an ultra-rapid data assimilation (URDA) method for adapting a given ensemble forecast for some particular variable of a dynamical system to given observation data which become available after the standard data assimilation and forecasting steps. Initial ideas have been suggested and tested by Etherthon 2006 and Madaus and Hakim 2015 in the framework of numerical weather prediction. The methods are, however, much more universally applicable to general non-linear dynamical systems as they arise in neuroscience, biology and medicine as well as numerical weather prediction. Here we provide a full analysis in the linear case, we formulate and analyse an ultra-rapid ensemble smoother and test the ideas on the Lorentz 63 dynamical system. In particular, we study the assimilation and preemptive forecasting step of an ultra-rapid data assimilation in comparison to a full ensemble data assimilation step as calculated by an ensemble Kalman square root filter. We show that for linear systems and observation operators, the ultra-rapid assimilation and forecasting is equivalent to a full ensemble Kalman filter step. For nonlinear systems this is no longer the case. However, we show that we obtain good results even when rather strong nonlinearities are part of the time interval $[t_0, t_n]$ under consideration. Then, an ultra-rapid ensemble Kalman smoother is formulated and numerically tested. We show that when the numerical model under consideration is different from the true model, used to generate the nature run and observations, errors in the correlations will also lead to errors in the smoother analysis. The numerical study is based on the popular Lorenz 1963 model system used in geophysics and life sciences. We investigate both the situation where the full system forecast is calculated and the situation important to practical applications where we study reduced data, when only one or two variables are known to the URDA scheme.